# notes - NIM Coins are arranged in columns two players...

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MATH 134 - Math and Politics NIM Coins are arranged in columns; two players alternate turns; the player who takes the last coin wins. Game tree- organized way to look at all possible configurations. o Nodes- configurations. o Edges- possible moves. o Branch- path from start to leaf (possible game). To determine who has a winning strategy, use Backwards Induction/Pruning/Dynamic Programming . o Look at nodes that have only leaves below it. We assume that at every stage, the player will act rationally, in his best interest. Dollar Auction Invented by Shubik in 1971 as a model for escalation (ex: Cold War) and solved by O’Neil in 1986 (for conservative convention. 2 players bid for same prize; on 1 st turn player 1 must bid, after players can bid or pass (game ends); both players’ bid goes to auctioneer; s=stakes; b=bankroll. Goal: to make biggest profit or smallest loss. Punishing convention - pick worst outcome for opponent (real life ex: poker). Conservative convention - minimize risk by picking lowest bid (real life ex: regular auction). O’Neil’s Theorem - if bankroll is b>0\$ and stakes are s>1 and the conservative convention is followed, then there is an optimal 1 st bid for P1 such that after that bid, P2 will pass. It can be determined by: take bankroll b and repeatedly subtract s-1 from b. The last value obtained before reaching zero or a negative # is the optimal bid. For problems, would need a bid #, stake #, and would have to show a game tree showing all possible results, then need to label win loss total with (+/-x,+/-y). After, need to show either punishing or conservative convention, as well as backwards induction. 2x2 Ordinal Games 2 players, row and column, & each has two alternatives to choose between (cooperate [C] and noncooperate [N]); play consists of N; each player ranks outcome (out of 4 possible) according to preference, 4 being best & 1 being worst (ranking-ordinal). A game in which row and column prefer to cooperate is called a

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notes - NIM Coins are arranged in columns two players...

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